Radar Systems -Unit- I : Radar Equation

RADAR SYSTEMS
B.TECH (IV YEAR – I SEM)
Prepared by:
Mr. P.Venkata Ratnam.,M.Tech.,(Ph.D)
Associate Professor
Department of Electronics and Communication Engineering
RAJAMAHENDRI INSTITUTE OF ENGINEERING & TECHNOLOGY
(Affiliated to JNTUK, Kakinada, Approved by AICTE - Accredited by NAAC )
Bhoopalapatnam, Rajamahendravaram, E.G.Dt, Andhra Pradesh

Course Content of Unit- I : Radar Equation
Modified Radar Range Equation
SNR
Probability of detection
Probability of False Alarm,
Integration of Radar Pulses RCS of Targets
Creeping Wave
Transmitter Power
PRF and Range Ambiguities
System Losses (qualitative treatment)
Illustrative Problems.

The simple form of radar range equation is
We know the following relation between Gain of
directional antenna and Effective Aperture Ae
Modified Radar Range Equation

Substitute this equation in above, We get
This is the one form of modified radar range equation

The noise figure Fn of a receiver is defined by the
equation:
Fn = No/ k To Bn Ga and
Rearranging the above two equations for Fn , the
input signal may be expressed as

If the minimum detectable signal Pmin or Smin is that
value of Si corresponding to the minimum ratio of
output (IF) signal-to-noise ratio (So/No )min necessary
for detection, then

Signal to Noise Ratio :
The signal to noise ratio, SNR or S/N ratio is one of
the most straightforward methods of measuring radio
receiver sensitivity.
It defines the difference in level between the signal and
the noise for a given signal level.
The lower the noise generated by the receiver, the
better the signal to noise ratio.
The signal-to-noise ratio at the output of the IF
amplifier necessary to achieve a specified probability of
detection without exceeding a specified probability of
false alarm.

 Any noise introduced by the first RF amplifier will be
added to the signal and amplified by subsequent
amplifiers in the receiver.
 As the noise introduced by the first RF amplifier will be
amplified the most, this RF amplifier becomes the most
critical in terms of radio receiver sensitivity
performance.
 Thus the first amplifier of any radio receiver should be a
low noise amplifier.

 The difference is normally shown as a ratio between
the signal and the noise, S/N, and it is normally
expressed in decibels.
 As the signal input level obviously has an effect on this
ratio, the input signal level must be given. This is
usually expressed in microvolts.
 The first is the actual bandwidth of the receiver. As the
noise spreads out over all frequencies it is found that
the wider the bandwidth of the receiver, the greater
the level of the noise.

Typically a certain input level required to give a 10 dB
signal to noise ratio is specified.
The signal to noise ratio is the ratio between the
wanted signal and the unwanted background noise. It
can be expressed in its most basic form using the S/N
ratio formula below:

Probability of detection :
A simplified block diagram of a radar receiver that
employs an envelope detector followed by a threshold
decision is shown in Fig
IF amplifier with bandwidth BIF followed by a second
detector and a video amplifier with bandwidth BV
The second detector and video amplifier are assumed to
form an envelope detector, that is, one which rejects the
carrier frequency but passes the modulation envelope.

To determine the Probability of false alarm when noise
alone is assumed to be present as input to the receiver.
The noise entering the IF amplifier is assumed to be
Gaussian, with probability-density function given by

Compare this with the Standard Probability density
function of Gaussian noise
If Gaussian noise were passed through a narrowband
IF filter whose Bandwidth is small compared with its
mid band frequency

The probability density of the envelope of the noise
voltage output is shown by the form of Rayleigh
probability-density function
Where R is the amplitude of the envelope of the filter
output. The probability that the envelope of the noise
voltage will lie between the values of V1 and V2 is

Course Content of Unit- I : Radar Equation
Modified Radar Range Equation
SNR
Probability of detection
Probability of False Alarm
Integration of Radar Pulses RCS of Targets
Creeping Wave
Transmitter Power
PRF and Range Ambiguities
System Losses (qualitative treatment)
Illustrative Problems.

Probability of False Alarm :
The probability that the noise voltage envelope will
exceed the voltage threshold VT is
Whenever the voltage envelope exceeds the threshold
VT, a target is considered to have been detected.
Since the probability of a false alarm is the probability
that noise will cross the threshold, the above equation
gives the probability of a false alarm, denoted by Pfa.

 The probability of false alarm as given above by itself
does not indicate that Radar is troubled by the false
indications of Target.
 The time between the false alarms TFA is a better
measure of the effect of Noise on the Radar
performance.

The average time interval between crossings of the
threshold by noise alone is defined as the false-
alarm time TFA
The false-alarm probability may also be defined as
the ratio of the duration of time the envelope is
actually above the threshold to the total time it
could have been above the threshold, i.e.

 Where tK and TK are shown in the Figure above. The
average duration of a noise pulse is approximately the
reciprocal of the bandwidth B, which in the case of the
envelope detector is BIF.
 Average time between false alarms Tfa is directly
proportional to the Threshold to noise ratio and
inversely proportional to the Bandwidth.

Integration of Radar Pulses RCS of Targets :
The number of pulses are usually returned from any
target on each radar scan and can be used to improve
detection.
The number of pulses nB returned from a point target
as the radar antenna scans through its beam width is
nB = θB . fP / θ’S = θB . fP / 6 ωm
Where θB = antenna beam width, deg
fP = pulse repetition frequency, Hz
θ’S = antenna scanning rate, deg/s
ωm = antenna scan rate, rpm

The process of summing all the radar echo pulses for the
purpose of improving detection is called integration.
Integration may be accomplished in the radar receiver
either before the second detector (in the IF) or after the
second detector (in the video).
Integration before the detector is called pre detection or
coherent integration.
In this the phase of the echo signal is to be preserved if
full benefit is to be obtained from the summing process
Integration after the detector is called post detection or
non coherent integration.

In this phase information is destroyed by the second
detector. Hence post detection integration is not
concerned with preserving RF phase.
Due to this simplicity it is easier to implement in most
applications, but is not as efficient as pre detection
integration.
If n pulses, all of the same signal-to-noise ratio, were
integrated by an ideal pre detection integrator, the
resultant or integrated signal-to-noise (power) ratio
would be exactly n times that of a single pulse.

If the same n pulses were integrated by an ideal post
detection device, the resultant signal-to-noise ratio
would be less than n times that of a single pulse.
This loss in integration efficiency is caused by the
nonlinear action of the second detector, which
converts some of the signal energy to noise energy in
the rectification process.
Due to its simplicity, Post detection integration is
preferred many a times even though the integrated
signal-to-noise ratio may not be as high as that of Pre-
detection.

 The efficiency of post detection integration relative to ideal
pre-detection integration has been computed by Marcum
when all pulses are of equal amplitude.
 The integration efficiency may be defined as follows:
 Where n = Number of pulses integrated
(S/N )1 = Value of signal-to-noise ratio of a single pulse
required to produce a given probability of
detection (for n = 1)
(S/N )n = Value of signal-to-noise ratio per pulse required to
produce the same probability of detection when n
pulses ( of equal amplitude ) are integrated

The improvement in the signal-to-noise ratio when n
pulses are integrated post detection is n.Ei(n) and is the
integration-improvement factor.
It may also be thought of as the effective number of
pulses integrated by the post detection integrator.
Integration loss in decibels is defined as
Li(n) = 10 log [1/Ei(n)].
The radar equation with n pulses integrated can be
written

Radar Cross Section of Targets:
 The Radar Cross Section (σ ) is the property of a scattering
object which represents the magnitude of the echo signal
returned to the radar by the target.
 An object exposed to an electromagnetic wave disperses
incident energy in all directions.
 This spatial distribution of energy is called scattering, and
the object itself is often called a scatterer.
 The energy scattered back to the source of the wave (called
backscattering) constitutes the radar echo of the object.
 The intensity of the echo is described explicitly by the
Radar Cross Section of the object.

The formal definition of radar cross section is

The RCS depends on the characteristic dimensions of
the object compared to the radar wavelength.
When the wavelength is large compared with the
object dimensions, scattering is said to be in Rayleigh
region.
The RCS in Rayleigh region is proportional to the
fourth power of the frequency.
When wavelength is small compared with object
dimension is called Optical region.
In between Rayleigh region and optical region is the
Resonance region.( Approx. 1 to 10 wavelengths)

RCS of Simple Targets:
i)Sphere : A perfectly conducting sphere acts as isotropic
radiator, it scattered uniformly into all 4π steradians.
The figure below as a function of its circumference
measured in wavelengths.(2πa/λ where a is the radius
of the sphere and λ is the wavelength).

The plot consists of three regions
1. Rayleigh Region:
The region where the size of the sphere is small
compared with the wavelength (2πa/λ << 1) is called
the Rayleigh region.
2. Optical region:
It is at the other extreme from the Rayleigh region
where the dimensions of the sphere are large
compared with the wavelength (2πa/λ >> 1).
3.Mie or Resonance region:
Between the optical and the Rayleigh region is the
Mie, or resonance, region. The cross section is
oscillatory with frequency within this region

ii)Cone-sphere: The cross section of cone-sphere is very
low and is considered to be of ballistic missile.
A large cross section occurs when a radar view the cone
perpendicular to its surface, RCS as a function of 2πa/λ

Creeping Wave :
Creeping waves play an important role in the analysis
of electromagnetic scattering by large objects with
curved boundaries.
According to the principle of diffraction, when a wave
front passes an obstruction, it spreads out into the
shadowed space.
A creeping wave in electromagnetism or acoustics is
the wave that is diffracted around the shadowed
surface of a smooth body such as a sphere.
Creeping waves greatly extend the ground wave
propagation of long wavelength radio.
In radar ranging, the creeping wave return appears to
come from behind the target.

In sphere the RCS rises quickly from a value of zero to a
peak near 2πa/λ = 1 and then executes a series of
decaying undulations as the sphere becomes electrically
larger.
The undulations are due to two distinct contributions to
the echo, one a specular reflection from the front of the
sphere and the other a creeping wave that skirts the
shadowed side.
The undulations become weaker with increasing 2πa/λ
because the creeping wave loses more energy the longer
the electrical path traveled around the shadowed side.

Transmitter Power :
 The average radar power Pav: It is defined as the average
transmitter power over the pulse-repetition period.
 If the transmitted waveform is a train of rectangular pulses
of width τ and pulse-repetition period Tp = 1/fp , then the
average power is related to the peak power by
 The peak power: The power Pt in the radar equation is
called the peak power. This is not the instantaneous peak
power of a sine wave.
 It is the power averaged over that carrier-frequency cycle
which occurs at the maximum power of the pulse.

Duty cycle: The ratio Pav/Pt, τ/TP, or τ. fP is called the
duty cycle of the radar.
A pulse radar for detection of aircraft might have
typically a duty cycle of 0.001, while a CW radar which
transmits continuously has a duty cycle of unity.
Writing the radar equation in terms of the average
power rather than the peak power, we get
The bandwidth and the pulse width are grouped
together since the product of the two is usually of the
order of unity in most pulse-radar applications.

PRF and Range Ambiguities :
The pulse repetition frequency (PRF) is
determined primarily by the maximum range at
which targets are expected.
If the PRF is made too high, the likelihood of
obtaining target echoes from the wrong pulse
transmission is increased.
Echo signals received after an interval exceeding
the pulse-repetition period are called multiple
time around echoes.

Consider the three targets labeled A, B, and C in the
figure(a) below.
Target A is located within the maximum unambiguous
range Runamb [= C.TP /2] of the radar.
Target B is at a distance greater than Runamb but less than
2Runamb .
The target C is greater than 2Runamb but less than
3Runamb.
The appearance of the three targets on an A-scope is
shown in the figure (b)below.

 The multiple-time-around echoes on the A-scope cannot be
distinguished from proper target echoes actually within the
maximum unambiguous range.
 Only the range measured for target A is correct; those for B
and C are not.
 Echoes from multiple-time-around targets will be spread
over a finite range as shown in the figure (c) below.
 The number of separate pulse repetition frequencies will
depend upon the degree of the multiple time around
targets.
 Second-time targets need only two separate repetition
frequencies in order to be resolved.

Let if PRF has unambiguous range Runamb1 and range
corresponds to it is R1,Then the true range is given by
Rtrue = R1 or
Rtrue = R1 + Runamb1 or
Rtrue = R1 + 2Runamb1
The true range can be any of the above equations. To
find the correct range(Rtrue),PRF is changed to PRF2
with range 2Runamb2 and the apparent range is given by
Rtrue = R2 or
Rtrue = R1 + Runamb2 or
Rtrue = R1 + 2Runamb2

System Losses:
 The losses with in the radar system called system losses.
 The losses in a radar system reduce the signal-to-noise ratio
at the receiver output.
 The major source of losses are
Microwave plumbing losses
Antenna losses
Collapsing losses
Equipment degradation losses
Signal processing losses

Microwave plumbing losses :
 Plumbing losses are Transmission and Duplexer
losses.
This is loss in the transmission lines which connects
the transmitter output to the antenna. (Cables and
waveguides).
At the lower radar frequencies the transmission line
introduces little loss, unless its length is exceptionally
long.
At higher radar frequencies, loss/attenuation will not
be small and has to be taken into account.

Duplexer loss: The signal suffers attenuation as it
passes through the duplexer.
Generally, the greater the isolation required from the
duplexer on transmission, the larger will be the
insertion loss.
Insertion loss means the loss introduced when the
component is inserted into the transmission line.
For a typical duplexer it might be of the order of 1 dB.

Antenna losses :
The Antenna losses are Beam-shape, Scanning,
Radome, Phased array losses.
Beam-shape loss: The train of pulses returned
from the target to a scanning radar are modulated
in amplitude by the shape of the antenna beam
A beam shape loss accounts for the fact that the
maximum gain is used in the radar equation rather
than a gain which changes from pulse to pulse.

To properly take into account the pulse- train
modulation caused by the beam shape, the
computations of the probability of detection would
have to be performed assuming a modulated train of
pulses rather than constant-amplitude pulses.
But since this computation is difficult, a beam-shape
loss is added to the radar equation and a maximum
gain is employed in the radar equation rather than a
gain that changes pulse to pulse.

Let the one way power pattern be approximated by a
Gaussian shape.

Scanning loss: When the antenna scans rapidly
enough, the gain on transmit is not the same as the
gain on receive.
An additional loss has to be computed, called the
scanning loss.
The technique for computing scanning loss is similar
in principle to that for computing beam-shape loss.
Scanning loss is important for rapid-scan antennas or
for very long range radars such as those designed to
view extraterrestrial objects.

 Collapsing loss: If the radar were to integrate additional
noise samples along with the wanted Signal-to- noise
pulses, the added noise results in degradation called the
collapsing loss.
 Equipment losses: The transmitter power in the radar
equation was assumed to be the specified output power
 However, all transmitting tubes are not uniform in quality,
and even any individual tube performance will not be same
throughout its useful life.
 Also, the power is not uniform over the operating band of
frequencies.
 Thus, for one reason or another, the transmitted power may
be other than the design value.

Operator loss: An alert, motivated, and well-trained
operator performs as described by theory.
However, when distracted, tired, overloaded, or not
properly trained, operator performance will decrease.
The resulting loss in system performance is called
operator loss.
Field degradation: When a radar system is operated
under laboratory conditions by engineering personnel
and experienced technicians, the above mentioned
losses give a realistic description of the performance of
the radar.

However, when a radar is operated under field
conditions the performance usually deteriorates even
more than that can be accounted for by the above
losses.
To minimize field degradation Radars should be
designed with built-in automatic performance-
monitoring equipment.
 Careful observation of performance-monitoring
instruments and timely preventative maintenance will
minimize field degradation.

Radar Systems -Unit- I : Radar Equation

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